introduction to stochastic combinatorial optimizationintroduction to stochastic combinatorial...

Introduction to Stochastic Combinatorial Optimization

Introduction toStochastic Combinatorial Optimization

Stefanie KosuchPostDok at TCSLab

www.kosuch.eu/stefanie/

Guest Lecture at the CUGS PhD course”Heuristic Algorithms for Combinatorial Optimization

Problems”

Stefanie Kosuch Introduction to Stochastic Combinatorial Optimization 1/53

www.kosuch.eu/stefanie/


What is the interest of Stochastic Combinatorial Optimization?

Combinatorial ”real world problems” often subject to uncertainties

Not all parameters known when decision has to be made:market fluctuations, available capacity...

Own decision depends on future decision of other parties:competition, clients, government...

Setting of problem might change:weather, location...


What is the interest of Stochastic Combinatorial Optimization?

Combinatorial ”real world problems” often subject to uncertainties

Not all parameters known when decision has to be made:market fluctuations, available capacity...

Own decision depends on future decision of other parties:competition, clients, government...

Setting of problem might change:weather, location...

2011-01-28


• Random parameters easy to implement → Random variables• ”Code” other uncertainties in parameters


Definition

Stochastic Combinatorial Optimization concerns the study andresolution of Combinatorial Optimization problems that involveuncertainties.


Definition

Stochastic Combinatorial Optimization concerns the study andresolution of Combinatorial Optimization problems that involveuncertainties.

2011-01-28


Note that there is a slight difference between the usage of the term”Stochastic Optimisation” and ”Stochastic Programming”.Stochastic Programming designs the modeling and study ofoptimization problems that involve uncertainties.Stochastic Optimization addresses the study of optimization algorithmsthat are either randomized or created to solve stochastic programmingproblems.

However, these definitions are not always properly used and of course

both fields intersect in a lot of aspects.


Objectives of this lecture

Give you examples of SCO-problems.

Give you an idea of how uncertainties can be modeled (mostcommon models).

Give you an idea of why Stochastic Optimization is hard.

Give you an idea of how SCO-problems can be solved.

Give you an idea of why metaheuristics are important tools inSCO.



2 examples of SCO problems

Outline

1 2 examples of SCO problems2 Modeling Stochastic Combinatorial Optimization Problems

”Where” does the randomness occur?Randomness occurs in the objective function f .Randomness occurs in the constraint function g .

When are the actual parameters revealed?Parameters are revealed after decision has been made.Parameters are revealed before corrective decision is made.Parameters are revealed in several stages.

3 Solving Stochastic Combinatorial Optimization problemsProblems/DifficultiesDeterministic ReformulationSample Average ApproximationMetaheuristics for SCO problems

4 ConclusionFurther ReadingStefanie Kosuch Introduction to Stochastic Combinatorial Optimization 5/53



Deterministic Knapsack problem



2011-01-28




Deterministic knapsack problem: The problem consists in choosing a

subset out of a given set of items such that the total weight (or size) of

the subset does not exceed a given limit (the capacity of the knapsack)

and the total benefit/reward of the subset is maximized.



Stochastic Knapsack problem

$4-8 12-15 kg

$2-6 2-3 kg

$1-2 1-4 kg

$2-5 1-3 kg

$10-15 4-7 kg

?15 kg



$4-8 12-15 kg

$2-6 2-3 kg

$1-2 1-4 kg

$2-5 1-3 kg

$10-15 4-7 kg

?15 kg

2011-01-28




What happens if item rewards or weights are random? What is a feasible

solution? For example, is it allowed to add all items apart from the green

one although they might violate the capacity constraint? And what

happens if they do?



Possible ways to handle capacity constraint

knapsack constraint violated ⇒ penalty

probability of capacity violation restricted

decision can be corrected later (add. costs/reduced rewards)






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In the first two examples violation was acceptable. But what if aviolation is not allowed, in any case? Well, we could force our solution toalways respect the knapsack constraint. In this case at most 3 itemscould be chosen, at a much lower reward.

Or we could see, if a correction might be possible later (3rd example), i.e.

we chose the 4 items (not the green one) and then, if their total weight

exceeds the capacity, we reject one item.



Deterministic Graph Coloring



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Deterministic Graph Coloring: Color a graph such that no two adjacentvertices are colored in the same color and such that a minimum numberof colors is used.In this example: Use of 4 colors is optimal as graph contains completegraph with 4 vertices.

Application: Assignment problems. The vertices could represent

university courses, two courses are linked iff there is at least one student

that wants to attend both courses. Coloring the obtained modelgraph

with the minimum number of colors tells you how many time slots you

need to schedule these courses.



Stochastic Graph Coloring



2011-01-28




What if, at the moment where you have to create the schedule, you do

not know the decision of the students yet? And what if there are two

courses with a very low probability that a student wants to take both of

them? And what if you are running out of time slots? You might

consider coloring both vertices with the same color and reduce the

number of used colors:



Changing settings

set of edges random

set of vertices random

Changing parameters

allowed number of colors random

”cost” of colors random


Changing settings

set of edges random

set of vertices random

Changing parameters

allowed number of colors random

”cost” of colors random

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Vertex set random: You do not know which courses will take place in theend. For example a course might be cancelled due to lack of students.

Number of allowed colors random: The university might assign you a

restricted number of time slots, that might change in the future due to

changings in other programs.


Modeling

Outline







Modeling

Deterministic CO Model → Stochastic CO Model

maxx∈0,1n

f (x)

s.t. g(x) ≤ 0 →

minx∈0,1n

f (x , χ)

s.t. g(x , χ) ≤ 0

χ ∈ Ω ⊆ Rs : vector with random entries


Deterministic CO Model → Stochastic CO Model

maxx∈0,1n

f (x)

s.t. g(x) ≤ 0 →

minx∈0,1n

f (x , χ)

s.t. g(x , χ) ≤ 0

χ ∈ Ω ⊆ Rs : vector with random entries

2011-01-28


Modeling Stochastic Combinatorial Optimization Problems

If you have an SCO optimization with random parameters and you fixthese parameters, you get a deterministic CO problem.

The other way round, if you have a deterministic CO problem and you

assume some of the parameters to be ranom your problem gets

stochastic. Question: Where does the randomness occurs? Only in the

objective, only in the constraint, in both?


Modeling

”Where” does the randomness occur?

Outline







Modeling


Minimize an expectation

minx∈0,1n

E [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:

Good result ”on average”

Objective function can often be reformulated deterministically

Disadvantages:

We might encounter very ”bad cases”


Minimize an expectation

minx∈0,1n

E [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:

Good result ”on average”

Objective function can often be reformulated deterministically

Disadvantages:

We might encounter very ”bad cases”

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• E[X ]: expectation of random variable X


Modeling


Minimize variance

minx∈0,1n

Var [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:

Outcome more concentrated around mean

Possibility to reduce risk

Disadvantages:

Makes not much sense without benchmark for expected costs


Minimize variance

minx∈0,1n

Var [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:



Disadvantages:

Makes not much sense without benchmark for expected costs

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• Var [X ]: variance of random variable X


Modeling


Minimize variance

minx∈0,1n

λVar [f (x , χ)] + E [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:




Minimize variance

minx∈0,1n

λVar [f (x , χ)] + E [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:



2011-01-28




Role of λ: Control relative importance of expectation and variance in

your model.


Modeling


Minimize variance

minx∈0,1n

Var [f (x , χ)] λ·E [f (x , χ)]

s.t. g(x) ≤ 0

Advantages:





Modeling


Robust optimization

minx∈0,1n

maxχ∈Ω

f (x , χ)

s.t. g(x) ≤ 0

Advantages:

Worst case not too bad: Solution is robust

Disadvantages:

f (x , ·) needs to be bounded from above

Worst case might be very improbable

Average might be high


Robust optimization

minx∈0,1n

maxχ∈Ω

f (x , χ)

s.t. g(x) ≤ 0

Advantages:

Worst case not too bad: Solution is robust

Disadvantages:

f (x , ·) needs to be bounded from above

Worst case might be very improbable

Average might be high2011-01-28




• Robust Optmization generally not considered as being part ofStochastic Optmization, e.g. as the solution algorithms andapproaches are generally quite different.

• However, I think the presented worst case model can be of good usein many cases, at least as a subproblem.

• Most common assumed distribution: χi (uniformly) distributed overcertain interval


Modeling


Outline







Modeling


Worst case model

minx∈0,1n

f (x)

s.t. g(x , χ) ≤ 0 ∀χ ∈ Ω

Advantages:

Absolutely robust solution

Disadvantages:

Problem often infeasible or has only trivial solutions

Solution at high costs

Constraint forced to be satisfied in even very improbable cases


Worst case model

minx∈0,1n

f (x)

s.t. g(x , χ) ≤ 0 ∀χ ∈ Ω

Advantages:

Absolutely robust solution

Disadvantages:

Problem often infeasible or has only trivial solutions

Solution at high costs

Constraint forced to be satisfied in even very improbable cases2011-01-28




• consider a knapsack problem with no upper bound on random itemweights: only feasible solution would be to add no item at all.

• Worst case problem considered as robust optmization problem.


Modeling


Chance-Constrained model

minx∈0,1n

f (x)

s.t. P∃i : gi (x , χ) > 0 ≤ α

Advantages:

Very improbable cases can be ignored

Cost can be reduced

Disadvantages:

No restriction of ”magnitude” of allowed violation

What happens if constraint is violated?


Chance-Constrained model

minx∈0,1n

f (x)

s.t. P∃i : gi (x , χ) > 0 ≤ α

Advantages:

Very improbable cases can be ignored

Cost can be reduced

Disadvantages:

No restriction of ”magnitude” of allowed violation

What happens if constraint is violated?2011-01-28




PA: probability that event A occurs

P∃i : gi (x , χ) > 0 = 1− Pgi (x , χ) ≤ 0 ∀i

P∃i : gi (x , χ) > 0 ≤ α⇔ Pgi (x , χ) ≤ 0 ∀i ≥ 1− α


Modeling


Simple-Recourse model

minx∈0,1n1

f (x) +m∑i=1

di · E[[gi (x , χ)]+

]Advantages:

Costs in case of violation taken into account

”Magnitude” of violation can be controlled

Disadvantages:

Probability of violation not restricted


Simple-Recourse model

minx∈0,1n1

f (x) +m∑i=1

di · E[[gi (x , χ)]+

]Advantages:

Costs in case of violation taken into account

”Magnitude” of violation can be controlled

Disadvantages:

Probability of violation not restricted

2011-01-28




• di > 0 penalty per ”unit of expected amount of violation”• [x ]+ = max(0, x)• [gi (x , χ)]+ = 0 iff constraint i satisfied, [gi (x , χ)]+ gives ”amount”

of violation otherwise• E [[gi (x , χ)]+]: expected amount of violation• Combine Simple Recourse and Chance constraint in order to control

both the magnitude and probability of violation


Modeling


Two-Stage model

minx∈0,1n1

f (x) + E [Q(x , χ)]

s.t. Q(x , χ) = miny∈0,1n2

f (y)

s.t. g(x , y , χ) ≤ 0

Advantages:

Violation of constraint not permitted

Corrections in case of violation taken into account

Disadvantages:

Problem extremely hard to solve:

→ Non-convex, non-continuous objective function→ No closed-form expression of objective function→ Second-stage problem NP-hard


Two-Stage model

minx∈0,1n1

f (x) + E [Q(x , χ)]

s.t. Q(x , χ) = miny∈0,1n2

f (y)

s.t. g(x , y , χ) ≤ 0

Advantages:

Violation of constraint not permitted

Corrections in case of violation taken into account

Disadvantages:

Problem extremely hard to solve:

→ Non-convex, non-continuous objective function→ No closed-form expression of objective function→ Second-stage problem NP-hard

2011-01-28




More general: First stage can of course have additional constraints.


Modeling


Deterministic Knapsack Problem

maxx∈0,1n

n∑i=1

rixi

s.t.n∑

i=1

wixi ≤ c

Simple Recourse Knapsack Problem

maxx∈0,1n

n∑i=1

rixi − d · E

[[

n∑i=1

χixi − c]+

]


Deterministic Knapsack Problem

maxx∈0,1n

n∑i=1

rixi

s.t.n∑

i=1

wixi ≤ c


maxx∈0,1n

n∑i=1

rixi − d · E

[[

n∑i=1

χixi − c]+

]

2011-01-28




• [∑n

i=1 χixi − c]+ = overweight• E

[[∑n

i=1 χixi − c]+]: expected overweight

• d > 0 penalty per overweight unit


Modeling


Two-Stage Knapsack Problem

(TSKP) maxx∈0,1n

n∑i=1

rixi + E[Q(x , χ)]

s.t. Q(x , χ) = maxy+,y−∈0,1n

n∑i=1

r iy+i −

n∑i=1

diy−i ,

s.t. y+j ≤ 1− xj , j = 1, . . . , n,

y−j ≤ xj , j = 1, . . . , n,n∑

i=1

(xi + y+i − y−i )χi ≤ c .



(TSKP) maxx∈0,1n

n∑i=1

rixi + E[Q(x , χ)]

s.t. Q(x , χ) = maxy+,y−∈0,1n

n∑i=1

r iy+i −

n∑i=1

diy−i ,

s.t. y+j ≤ 1− xj , j = 1, . . . , n,

y−j ≤ xj , j = 1, . . . , n,n∑

i=1

(xi + y+i − y−i )χi ≤ c .

2011-01-28




• Items can be added and/or removed in the second stage• In the end remaining items need to respect knapsack capacity• x : decision vector of 1st stage• y+, y−: decision vectors of 2nd stage (recourse action)• r i < ri , di > ri• If r i ≥ ri : Add item i in 2. stage• If di ≤ ri : Add item in 1. stage (removal is without cost or one even

gains)


Modeling

When are the actual parameters revealed?

Outline







Modeling


Static Stochastic Optimization problems

Random parameters revealed after decision has been made.

For decision maker parameters are revealed ”once for all”.

No corrective decision can be made.

Minimize expectation and/or variance

Robust/Worst Case Optimization

Chance-Constrained Optimization

Simple Recourse Model


Static Stochastic Optimization problems

Random parameters revealed after decision has been made.


No corrective decision can be made.

Minimize expectation and/or variance

Robust/Worst Case Optimization

Chance-Constrained Optimization


2011-01-28




• Simple Recourse model: is paying a penalty a corrective decision?No.


Modeling


Two-Stage Optimization problems


Random parameters revealed after first-stage decision hasbeen made.

Corrective decision can be made once the parameters areknown.

Two-Stage Model



Two-Stage Optimization problems


Random parameters revealed after first-stage decision hasbeen made.

Corrective decision can be made once the parameters areknown.

Two-Stage Model


2011-01-28




• Simple Recourse model: can be reformulated as Two-Stage decisionmodel

• Continuous second stage decision variables yi serve to ”correctconstraints”

• One variable for each constraint• Second stage constraints: gi (x , χ) ≤ 0 + yi• Optimal second-stage decision: yi = [gi (x , χ)]+


Modeling


Multi-Stage Optimization problems

Parameters are revealed in several stages.

Corrective decision can be made in each stage.

Which parameters are revealed in which stage generallydefined.

Decisions do only depend on already revealed parameters.

Two-Stage Model

Multi-Stage Model


Multi-Stage Optimization problems

Parameters are revealed in several stages.

Corrective decision can be made in each stage.

Which parameters are revealed in which stage generallydefined.

Decisions do only depend on already revealed parameters.

Two-Stage Model

Multi-Stage Model

2011-01-28




• Attention: In deterministic multi-period decision problems: futureparameters all knowndecisions based on current parameters and future changings⇒ multi-period decision problems problems are deterministicproblems


Solving Stochastic Combinatorial Optimization problems

Outline








Problems/Difficulties

Outline









Structural Difficulties

Non-convexity

Non-continuous objective functions

No closed-form expression for objective function

Two-Stage model

minx∈0,1n1

f (x) + E [Q(x , χ)]

s.t. Q(x , χ) = miny∈0,1n2

f (y)

s.t. g(x , y , χ) ≤ 0


Structural Difficulties

Non-convexity

Non-continuous objective functions

No closed-form expression for objective function

Two-Stage model

minx∈0,1n1

f (x) + E [Q(x , χ)]

s.t. Q(x , χ) = miny∈0,1n2

f (y)

s.t. g(x , y , χ) ≤ 02011-01-28




The Two-Stage model is an extrem example for the structural difficulties

as generally all of them are present (at least in case of second stage

integer or binary decision variables)




Computational Difficulties

Expectations and probabilities = (multi-dimensional) integrals!

Evaluating objective function might be NP-hard

High number of binary decision variables and constraints






(TSKP) maxx∈0,1n

n∑i=1

rixi + E[Q(x , χ)]

s.t. Q(x , χ) = maxy+,y−∈0,1n

n∑i=1

r iy+i −

n∑i=1

diy−i ,

s.t. y+j ≤ 1− xj , j = 1, . . . , n,

y−j ≤ xj , j = 1, . . . , n,n∑

i=1

(xi + y+i − y−i )χi ≤ c .



(TSKP) maxx∈0,1n

n∑i=1

rixi + E[Q(x , χ)]

s.t. Q(x , χ) = maxy+,y−∈0,1n

n∑i=1

r iy+i −

n∑i=1

diy−i ,

s.t. y+j ≤ 1− xj , j = 1, . . . , n,

y−j ≤ xj , j = 1, . . . , n,n∑

i=1

(xi + y+i − y−i )χi ≤ c .

2011-01-28




An example for NP-hardness of the second-stage problem is the

Two-Stage Knapsack problem, as the second stage problem can be

shown to be a ”simple” knapsack problem.



Deterministic Reformulation

Outline









Idea

→ Reformulate problem as a deterministic optimization problem

→ Use already existing solvers to solve obtained problem

→ Adapt existing algorithms to the special structure of theobtained problem

Problem

Generally only possible under assumption of special distributions!


Idea

→ Reformulate problem as a deterministic optimization problem

→ Use already existing solvers to solve obtained problem

→ Adapt existing algorithms to the special structure of theobtained problem

Problem

Generally only possible under assumption of special distributions!

2011-01-28




Currently the most practiced approach to solve stochastic compbinatorial

optimization problem. Unfortunately, as I think that if we tried to be a

bit more innovative concerning the creation of special algorithms for

SCO, we might advance faster.




Ex. 1: Chance-constrained Knapsack prob. (normal distribution)

maxx∈0,1n

n∑i=1

rixi

s.t. Pn∑

i=1

χixi > c ≤ α

Assume:

χ ∼ N (µ,Σ) and α < 0.5⇒

Pn∑

i=1

χixi > c ≤ α⇔n∑

i=1

xiµi + Φ−1(1− α)‖Σ1/2x‖ ≤ c︸︷︷︸

Second Order Cone Constraint!


Ex. 1: Chance-constrained Knapsack prob. (normal distribution)

maxx∈0,1n

n∑i=1

rixi

s.t. Pn∑

i=1

χixi > c ≤ α

Assume:

χ ∼ N (µ,Σ) and α < 0.5⇒

Pn∑

i=1

χixi > c ≤ α⇔n∑

i=1

xiµi + Φ−1(1− α)‖Σ1/2x‖ ≤ c︸︷︷︸

Second Order Cone Constraint!

2011-01-28




• N (µ,Σ): joint probability distribution for random vector χ• N (·, ·): (joint) normal distribution• µ: vector of expectations of components of χ: µi = E[χi ]• Σ: n × n covariance matrix• here: Σ diagonal as weights assumed independently distributed• Φ: cumulative distribution function of standard normal distribution• values of Φ−1 can be looked up in tables• α < 0.5 needed for Φ−1(1− α) to be positive• otherwise constraint not convex• obtained constraint can be evaluated (one can check easily for

feasibility)• obtained problem type (Second Order Cone Problem) has been

studied a lot and special algorithms have been proposed → no more”miracle” about how to solve the chance-constraint knapsackproblem




Ex. 2: Chance-constrained Knapsack prob. (discrete distribution)

maxx∈0,1n

n∑i=1

rixi s.t. Pn∑

i=1

χixi > c ≤ α

Assume:K outcomes χ1, . . . , χK with prob.’s p1, . . . , pK

Introduce:K binary decision variables z1, . . . , zK

Replace Chance-Constraint by

n∑i=1

χki xi ≤ c + Mzk ∀ k = 1, . . . ,K ,

K∑k=1

pkzk ≤ α


Ex. 2: Chance-constrained Knapsack prob. (discrete distribution)

maxx∈0,1n

n∑i=1

rixi s.t. Pn∑

i=1

χixi > c ≤ α

Assume:K outcomes χ1, . . . , χK with prob.’s p1, . . . , pK

Introduce:K binary decision variables z1, . . . , zK

Replace Chance-Constraint by

n∑i=1

χki xi ≤ c + Mzk ∀ k = 1, . . . ,K ,

K∑k=1

pkzk ≤ α2011-01-28




• M ≥ maxk(∑n

i=1 χki − c)

• zk = 1: scenario k is ”ignored”corresponding constraint

∑ni=1 χ

ki xi ≤ c + Mzk always satisfied

• total probability of ”ignored” scenarios must not exceed α



Sample Average Approximation

Outline









Idea of the SAA

→ Sample K outcomes for random parameters

→ Assign probability 1/K to each sample

→ Replace distribution by finite sample

Advantages

Approximate information about underlying distribution X

Approximate closed-form expression for objective function X

Approximate deterministic reformulation of problem X

Smaller number of scenarios X


Idea of the SAA




Advantages

Approximate information about underlying distribution X

Approximate closed-form expression for objective function X

Approximate deterministic reformulation of problem X

Smaller number of scenarios X2011-01-28




Note that SAA methods are not usable for robust optimization as inrobust optimization we are concerned about the worst case. This cannotbe reflected by working only on a sample.

Concerning ”Smaller number of scenarios”: Of course sample average

approximation can also be used in case of a discrete probability

distribution with a huge number of scenarios. If an approximation of the

solution is all we need, a SAA with smaller sample might be solved

instead.




Idea of the SAA




Disadvantages

Solution of SAA might be infeasible for original problem

Solution of SAA might be non-optimal for original problem

To approximate original problem high number of samplesmight be needed






maxx∈0,1n

n∑i=1

rixi − d · E

[[

n∑i=1

χixi − c]+

]

Sample Average Approximation:

maxx∈0,1n

n∑i=1

rixi − d ·K∑

k=1

1

K[

n∑i=1

χki xi − c]+




Metaheuristics for SCO problems

Outline









”Positive” Features

Basically same as for deterministic comb. opt.

Based on sample average approximations

Increase sample size to obtain convergence.

Additional diversification due to randomness of samples


”Positive” Features

Basically same as for deterministic comb. opt.

Based on sample average approximations

Increase sample size to obtain convergence.

Additional diversification due to randomness of samples

2011-01-28




Most metaheuristics for stochastic combinatorial optimisation work asfollows: At the beginning of each iteration you draw a sample of therandom parameters and create the corresponding SAA. The rest of theiteration this SAA is used to create new solutions, to compare theirquality etc. In the next iteration, a new, slightly bigger sample is drawnetc..

Once more this approach is not possible for robust optimization (see

remark in the SAA subsection).




Difficulties

Evaluation of objective function expensive

Comparison of quality of solutions difficult

SCO problems generally have a lot of local optima

Values of local optima can be close


Difficulties

Evaluation of objective function expensive

Comparison of quality of solutions difficult

SCO problems generally have a lot of local optima

Values of local optima can be close

2011-01-28




In general you have to evaluate the objective function many times when

applying a metaheuristic, in order to compare the quality of found

solutions.


Conclusion

Outline







Conclusion

Summary

Modeling SCO problems:

→ Randomness in objective function?→ Randomness in constraint?→ Violation of constraint possible? Penalty?→ Correction of decision possible? How often?

Solving SCO problems:

→ Deterministic equivalent formulation→ Approximation of problem using sampling→ Meta-Heuristics



Conclusion

Greatest Challenges in Stochastic Combinatorial Optimization

Modeling Combinatorial Real World problems withuncertainties as SCO problems

Solve ”realistic” sized problems in reasonable time

Find more adapted solution techniques (for generaldistributions)

Find efficient (Meta)Heuristics



Conclusion

Further Reading

Outline







Conclusion

Further Reading

Alexander Shapiro, Darinka Dentcheva, Andrzej RuszczynskiLectures on Stochastic Programming (2009)pdf available online

Andras PrekopaStochastic Programming (1995).Springer.

Anton J. Kleywegt, Alexander Shapiro, Tito Homem-de-MelloThe Sample Average Approximation Method for Stochastic DiscreteOptimization (2002)SIAM Journal on Optimization

Peter Kall, Stein W. WallaceStochastic Programming (1994)pdf available online

Stochastic Programming CommunityStochastic Programming Community Home Pagehttp://stoprog.org/

Leonora Bianchi, Marco Dorigo, Luca Maria Gambardella, Walter J. GutjahrMetaheuristics in Stochastic Combinatorial Optimization: a Survey (2006)pdf available online


http://stoprog.org/

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